Sampling Distributions for Sample Slopes

In simple linear regression, the population regression line is defined by the equation $y = \alpha + \beta x$, where $\beta$ represents the true, unknown rate of change in the response variable for every one-unit increase in the explanatory variable.

When we take a random sample of size $n$, we calculate a sample regression line $\hat{y} = a + bx$, where $b$ is the sample slope used to estimate the population parameter $\beta$.

Because different random samples yield different values of $b$, the sample slope is a random variable with its own probability distribution, known as the sampling distribution of the sample slope.

Center: The mean of the sampling distribution of $b$ is equal to the population slope, $\mu_b = \beta$. This indicates that the sample slope is an unbiased estimator of the population slope.

Spread: The theoretical standard deviation of the sample slope is $\sigma_b = \frac{\sigma}{\sigma_x \sqrt{n}}$, where $\sigma$ is the standard deviation of the population residuals and $\sigma_x$ is the standard deviation of the $x$-values.

Interpretation: The spread of the distribution decreases as the sample size $n$ increases or as the variability in the $x$-values increases, leading to more precise estimates of the slope.

In practice, we rarely know the population standard deviation of residuals ($\sigma$), so we must estimate it using the standard error of the residuals ($s$), calculated as $s = \sqrt{\frac{\sum(y_i - \hat{y}_i)^2}{n-2}}$.

The standard error of the slope ($s_b$) is the estimated standard deviation of the sampling distribution and is calculated as $s_b = \frac{s}{s_x \sqrt{n-1}}$.

This value $s_b$ describes how much the sample slope $b$ typically varies from the population slope $\beta$ in repeated random sampling.

When the conditions are met, the standardized statistic $t = \frac{b - \beta}{s_b}$ follows a t-distribution.

The degrees of freedom for this distribution are $df = n - 2$.

We use $n - 2$ because we have estimated two parameters from the sample data to create the regression line: the intercept ($a$) and the slope ($b$).

Identify the Parameter: Always define $\beta$ in the context of the problem (e.g., 'the true slope of the relationship between height and weight').

Check the df: Remember that for regression, $df = n - 2$. Using $n - 1$ is a common mistake that leads to incorrect critical values.

Residual Plot Interpretation: If a residual plot shows a clear curve, the linearity condition is violated, and the sampling distribution of the slope may not be valid.

Computer Output: On exams, you are often given a table. The sample slope $b$ is usually in the 'Coef' column next to the variable name, and $s_b$ is in the 'SE Coef' column.